Aberystwyth University
Fluid velocity based simulation of hydraulic fracture: a penny shapedmodel—part IPeck, Daniel; Wrobel, Michal; Perkowska, Monika; Mishuris, Gennady
Published in:Meccanica
DOI:10.1007/s11012-018-0899-y
Publication date:2018
Citation for published version (APA):Peck, D., Wrobel, M., Perkowska, M., & Mishuris, G. (2018). Fluid velocity based simulation of hydraulic fracture:a penny shaped model—part I: The numerical algorithm. Meccanica, 53(15), 3615-3635.https://doi.org/10.1007/s11012-018-0899-y
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Fluid velocity based simulation of hydraulic fracture:a penny shaped model—part I: the numerical algorithm
Daniel Peck . Michal Wrobel . Monika Perkowska . Gennady Mishuris
Received: 16 December 2016 / Accepted: 12 September 2018
� The Author(s) 2018
Abstract In the first part of this paper, a universal fluid
velocity based algorithm for simulating hydraulic frac-
ture with leak-off, previously demonstrated for the PKN
and KGD models, is extended to obtain solutions for a
penny-shaped crack. The numerical scheme is capable of
dealing with both the viscosity and toughness dominated
regimes, with the fracture being driven by a power-law
fluid. The computational approach utilizes two depen-
dent variables; the fracture aperture and the reduced fluid
velocity. The latter allows for the application of a local
condition of the Stefan type (the speed equation) to trace
the fracture front. The obtained numerical solutions are
carefully tested using various methods, and are shown to
achieve a high level of accuracy.
Keywords Penny-shaped crack � Hydraulicfracture �Universal algorithm � Power law fluid � Leak-off � Speed equation
1 Introduction
Hydraulic fracture (HF) is the phenomenon of a fluid
driven crack propagating in a solid material. It can be
encountered in various natural processes, such as
subglacial drainage of water or during the extension of
magmatic intrusions in the earth’s crust. Simultane-
ously the underlying physical mechanism is very
important in numerous man-made activities.
Hydrofracturing can appear as an unwanted and
detrimental factor in underground CO2 or waste
repositories [1]. On the other hand, intentionally
induced hydraulic fractures constitute the essence of
fracking technology - a method used when stimulating
unconventional hydrocarbon reservoirs [2] or for
geothermal energy exploitation [3]. All of these
applications create demand for a proper understanding
and prediction of the process of hydraulic fracture.
Asa result of themultiphysical nature of theunderlying
physical phenomenon and complex interactions between
thecomponentphysicalfields, themathematicalmodeling
of hydraulic fractures represents a significant challenge.
Themaindifficultiesarisedue to: (1) strongnon-linearities
resulting from interaction between the solid and fluid
phases, (2) singularities in the physical fields, (3) moving
boundaries, (4) degenerationof the governing equations at
the crack tip, (5) leak-off to the rock formation, (6)
pronounced multiscaling effects, vii) complex geometry.
The first theoretical models of hydraulic fracture
were created in 1950s (see for example [4] and [5]).
Electronic supplementary material The online version ofthis article (https://doi.org/10.1007/s11012-018-0899-y) con-tains supplementary material, which is available to authorizedusers.
D. Peck � G. Mishuris (&)
Aberystwyth University, Aberystwyth, UK
e-mail: [email protected]
M. Wrobel
AGH University of Science and Technology, Cracow,
Poland
M. Perkowska
EnginSoft S.p.A., Trento, Italy
123
Meccanica
https://doi.org/10.1007/s11012-018-0899-y(0123456789().,-volV)(0123456789().,-volV)
Subsequent research led to the formulation of the so-
called classic 1D models: PKN [6, 7], KGD (plane
strain) [8, 9] and penny-shaped/radial [10]. Up to the
1980s these very simplified models were used to
design and optimize the treatments used in HF. The
increasing number and size of fracking installations,
alongside the simultaneous advance in computational
techniques, resulted in the formulation of more
sophisticated and realistic models of HFs. A compre-
hensive review of the topic can be found in [11].
Though superseded in most practical applications,
the classic 1D models remain a significant avenue of
research into the fundamentals of HF. They enable one
to investigate some inherent features of the underlying
physical process, the mathematical structure of the
solution, and finally to construct and validate compu-
tational algorithms. Substantial advances have been
achieved in this area throughout the last 30 years by
way of a cyclical revision of these classic formulations.
It was not until 1985 [12] that the importance of the
solution tip asymptotics was first noticed, specifically
for the KGD and penny shaped cracks. The explicit
form of the tip asymptotic solution for the PKN model
was given in 1990 by Kemp [13]. Moreover, in this
publication the author remarked, for the first time, that
when properly posed the Nordgren’s model constitutes
a Stefan-type problem and as such needs an additional
boundary condition which equates the crack propaga-
tion speedwith the velocity of the fluid front. However,
this important ideawas abandoned for the next 20 years
until being rediscovered by Linkov [14] in 2011. The
author proved that the general HF problem is ill-posed
and proposed a regularization technique based on
application of the aforementioned Stefan condition—
called there the speed equation. The numerous inves-
tigations carried out since the beginning of the present
century for the KGD [15–18] and penny-shaped
models [19–21] have led to the importance of the
problem’s multiscale character being recognized. It is
now well understood that the global response of the
fluid driven fracture is critically dependent on the
interaction between competing physical processes at
various temporal and spatial scales. Depending on the
intensity of various energy dissipation mechanisms, as
well as the fracturing fluid and solid material proper-
ties, the hydraulic fracture evolves in the parametric
space encompassed by the limiting regimes: (1)
viscosity dominated, (2) toughness dominated, (3)
storage dominated, (4) leak-off dominated.
Bearing in mind the whole complexity of the
problem, it still remains an extremely challenging task
to deliver credible solutions which reflect all of the
desired features. The relative simplicity of the classic
1D models means that they are well suited to the task
of creating benchmarks, used when developing and
verifying more advanced solutions and algorithms.
For the KGD and PKN models one can find in the
literature a number of credible results, including
recently developed simple and accurate approximate
solutions, that can be utilized for the aforementioned
purposes [22–25]. A more complete review of recent
benchmarks will be provided in part II of this paper.
The aim of the first paper is to meet the demand for
benchmark solutions to the radial HF model and: (1) to
deliver a dedicated computational scheme capable of
obtaining highly accurate numerical solutions, (2) intro-
duce purely analytical solutions to the problem obtained
for a predefinednon-zero leak-off function, (3) introduce
and verify an alternative measure of the computational
error, to use when no analytical solution is available.
To this end the self-similar formulation of the
penny-shaped model will be analyzed. The numerical
computations will be performed according to a
modified form of the universal algorithm introduced
in [24, 25]. It employs a mechanism of fracture front
tracing, based on the speed equation approach [23],
coupled with an extensive use of information on the
crack tip asymptotics and regularization of the
Tikhonov type (the technical details of both concepts
can be found in [26, 27]). The modular architecture of
the computational scheme facilitates its adaptation to
the problem of radial HF.
It is worth stating that the second part of this paper
will introduce simple to use semi-analytical approx-
imations of numerical benchmark solutions obtained
for the case of an impermeable solid, and comparisons
with other available data will be performed. Both parts
are written in such a way that they can be read as
individual, independent papers (for a unified version
of the text, see arXiv:1612.03307).
The paper is organized as follows. The basic system
of equations describing the problem is given in
Sect. 2. Next, normalization to the dimensionless
form is carried out. In Sect. 3, comprehensive infor-
mation about the solution asymptotics is presented,
which is heavily utilized in the subsequent numerical
implementation. New computational variables, the
reduced fluid velocity and modified fluid pressure
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Meccanica
derivative, are introduced. The advantages of both are
outlined, and the problem is reformulated in terms of
the new variables. In Sect. 4 the governing system of
equations is reduced to the time independent self-
similar form. This formulation is used in Sect. 5 to
construct the computational algorithm. The accuracy
and efficiency of computations are examined against
newly introduced analytical benchmark examples.
Alternative error measures are proposed for the cases
where no closed-form analytical solution is available.
Section 6 contains the final discussion and conclu-
sions. Some additional information concerning the
limiting cases of Newtonian and perfectly plastic
fluids is collected in the appendices.
2 Problem formulation
Let us consider a 3D penny-shaped crack, defined in
polar coordinates by the system fr; h; zg, with asso-
ciated crack dimensions flðtÞ;wðtÞg as the fracture
radius and aperture respectively, noting that both are
functions of time. The crack is driven by a point source
located at the origin, which has a known pumping rate:
Q0ðtÞ. The fluid’s rheological properties are describedby the power-law model [28], as is common in the
literature. The rational behind this choice is outlined in
[25]. We have that, as the flow is axisymmetric, all
variables will be independent of the angle h and it is
sufficient to analyse the problem for only r� 0.
The fluid mass balance equation is as follows:
ow
otþ 1
r
o
orrqð Þ þ ql ¼ 0; 0\r\lðtÞ; ð1Þ
where qlðr; tÞ is the fluid leak-off function, represent-
ing the volumetric fluid loss to the rock formation in
the direction perpendicular to the crack surface per
unit length of the fracture. We will assume it to be a
predefined and smooth function which is bounded at
the fracture tip.
Meanwhile, q(r, t) is the fluid flow rate inside the
crack, given by the Poiseuille law:
qn ¼ �w2nþ1
M0op
or; ð2Þ
with p(r, t) being the net fluid pressure on the fracture
walls (i.e. p ¼ pf � r0, where pf is the total pressure
and r0 is the confining stress). The constant M0 is theso-called modified fluid consistency index
M0 ¼ 2nþ1ð2nþ 1Þn=nnM, where M stands for the
consistency index (relating the shear stress and strain
rate) and 0� n� 1 is the fluid behaviour index.
The non-local relationships between the fracture
aperture and the pressure (elasticity equations) are as
follows:
pðr; tÞ ¼ E0
lðtÞA½w�ðr; tÞ; wðr; tÞ ¼ lðtÞE0 A
�1½p�ðr; tÞ;
ð3Þ
where E0 ¼ Y=ð1� m2Þ, with Y being the Young’s
modulus and m the Poisson ratio. The operator A and
its inverse take the form:
A½w� ¼ �Z 1
0
owðglðtÞ; tÞog
Mr
lðtÞ ; g� �
dg; ð4Þ
A�1½p�¼8
p
Z 1
r=lðtÞ
nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2�ðr=lðtÞÞ2
qZ 1
0
gpðgnlðtÞ;tÞffiffiffiffiffiffiffiffiffiffiffiffi1�g2
p dgdn
�8
p
Z 1
0
gpðglðtÞ;tÞG r
lðtÞ;g� �
dg;
ð5Þ
with the pertinent kernels being:
M n; s½ � ¼ 1
2p
1
nK
s2
n2
� �þ n
s2 � n2E
s2
n2
� �; n[ s
s
s2 � n2E
n2
s2
� �; s[ n
8>><>>:
ð6Þ
Gðn; sÞ ¼
1
nF arcsin
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� n2
1� s2
s0@
1A���s2
n2
0@
1A; n[ s
1
sF arcsin
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� s2
1� n2
s !���n2
s2
!; s[ n
8>>>>>><>>>>>>:
ð7Þ
K, E are the complete elliptic integrals of the first and
second kinds respectively, and F is the incomplete
elliptic integral of the first kind, given in [29].
These equations are supplemented by the boundary
condition at r ¼ 0, which defines the intensity of the
fluid source, Q0:
limr!0
rqðr; tÞ ¼ Q0ðtÞ2p
; ð8Þ
the tip boundary conditions:
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Meccanica
wðlðtÞ; tÞ ¼ 0; qðlðtÞ; tÞ ¼ 0; ð9Þ
and appropriate initial conditions describing the
starting crack opening and length:
wðr; 0Þ ¼ w�ðrÞ; lð0Þ ¼ l0: ð10Þ
Additionally, it is assumed that the crack is in
continuous mobile equilibrium, and as such the
classical crack propagation criterion of linear elastic
fracture mechanics is imposed [30]:
KI ¼ KIc; ð11Þ
where KIc is the material toughness while KI is the
stress intensity factor. The latter is computed accord-
ing to the following formula [31]:
KIðtÞ ¼2ffiffiffiffiffiffiffiffiffiffiplðtÞ
pZ lðtÞ
0
rpðr; tÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2ðtÞ � r2
p dr: ð12Þ
Throughout this paper we accept the convention that
when KIc ¼ 0 the hydraulic fracture propagates in the
viscosity dominated regime. Otherwise the crack
evolves in the toughness dominated mode. Each of
these two regimes is associated with qualitatively
different tip asymptotics, which constitutes a singular
perturbation problem as KIc ! 0, and leads to serious
computational difficulties in the small toughness
range.
Finally, noting (1) and (8), the global fluid balance
equation is given by:
Z lðtÞ
0
r wðr; tÞ � w0ðrÞ½ � dr þZ t
0
Z lðtÞ
0
rqlðr; sÞ dr ds
¼ 1
2p
Z t
0
Q0ðsÞ ds: ð13Þ
The above set of equations and conditions represents
the typically considered formulation for a penny-
shaped hydraulic fracture [19].
In order to facilitate the analysis we shall utilize an
additional dependent variable, v, which describes the
average speed of fluid flow through the fracture cross-
section [23]. It will be referenced to in the text as the
fluid velocity (often referred to in the literature as the
particle velocity, e.g. [24, 25]), and is defined as:
vðr; tÞ ¼ qðr; tÞwðr; tÞ ; vnðr; tÞ ¼ � 1
M0 wnþ1 op
or: ð14Þ
We assume that the leak-off ql is such that the fluid
velocity is finite at the crack tip, meaning that v has the
following property:
limr!lðtÞ
vðr; tÞ ¼ v0ðtÞ\1: ð15Þ
Additionally, given that the fracture apex coincides
with the fluid front (there is no lag), and that the fluid
leak-off at the fracture tip is bounded, the so-called
speed equation [14] holds:
dl
dt¼ v0ðtÞ: ð16Þ
This Stefan-type boundary condition constitutes an
explicit method, as opposed to an implicit level-set
method [32], and can be effectively used to construct a
mechanism of fracture front tracing. The advantages
of implementing such a condition have been shown in
[24, 25].
2.1 Problem normalization
In order to make the presentation clearer, we will
assume that 0\n\1 in the main body of the text. Any
modification to the governing equations and numerical
scheme in the limiting cases n ¼ 0 and n ¼ 1 are
detailed in ‘‘Appendix 1’’.
We normalize the problem by introducing the
following dimensionless variables:
~r ¼ r
lðtÞ ;~t ¼ t
t1=nn
; ~wð~r; ~tÞ ¼ wðr; tÞl�
;
Lð~tÞ ¼ lðtÞl�
; ~qlð~r;~tÞ ¼t1=nn
l�qlðr; tÞ;
~qð~r;~tÞ ¼ t1=nn
l2�qðr; tÞ; ~Q0ð~tÞ ¼
t1=nn
l2�lðtÞQ0ðtÞ;
~vð~r; ~tÞ ¼ t1=nn
l�vðr; tÞ; ~pð~r; ~tÞ ¼ tn
M0 pðr; tÞ;
~KIc ¼1
E0 ffiffiffiffil�
p KIc; tn ¼M0
E0 ;
ð17Þ
where ~r 2 0; 1½ � and l� is chosen for convenience.
We note that such a normalization scheme has
previously been used in [24, 25], and that it is not
attributed to any particular influx regime or asymptotic
behaviour of the solution.
Under normalization scheme (17), the continuity
equation (1) can be rewritten in terms of the fluid
velocity (14) to obtain:
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Meccanica
o ~w
o~t� L0ð~tÞ
Lð~tÞ ~ro ~w
o~rþ 1
Lð~tÞ~ro
o~r~r ~w~vð Þ þ ~ql ¼ 0: ð18Þ
The fluid velocity (2) is expressed as:
~v ¼ � ~wnþ1
Lð~tÞo~p
o~r
� �1n
; ð19Þ
while the speed equation is now given by combining
(14)–(16):
~v0ð~tÞ ¼ L0ð~tÞ ¼ � ~wnþ1
Lð~tÞo~p
o~r
� �1n
~r¼1
\1: ð20Þ
The global fluid balance equation (13) is transformed
to:
Z 1
0
~r L2ð~tÞ ~wð~r; ~tÞ � L2ð0Þ ~w0ð~rÞ�
d~r
þZ ~t
0
Z 1
0
~rL2ðsÞ~qlð~r; sÞ d~r ds
¼ 1
2p
Z ~t
0
LðsÞ ~Q0ðsÞ ds:
ð21Þ
The notation for the elasticity Eqs. (3)–(5) takes the
form:
~pð~r; ~tÞ ¼ 1
Lð~tÞA½ ~w�ð~r; ~tÞ; ~wð~r; ~tÞ ¼ Lð~tÞA�1½~p�ð~r;~tÞ;
ð22Þ
where the operators denote:
A½ ~w�ð~r; ~tÞ ¼ �Z 1
0
o ~wðg;~tÞog
M ~r; g½ � dg; ð23Þ
A�1½~p�ð~r;~tÞ ¼ 8
p
Z 1
~r
nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 � ~r2
pZ 1
0
g~pðgn;~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� g2
p dg dn:
ð24Þ
From definition (12) and the fracture propagation
condition (11) we have that:
~KI ¼ ~KIc ¼2ffiffiffip
pffiffiffiffiffiffiffiffiLð~tÞ
q Z 1
0
~r~pð~r; ~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p d~r: ð25Þ
Note that through proper manipulation of (24) and the
use of (25), (22)2 can be expressed in the following
form:
~wð~r;~tÞ ¼ 8
pLð~tÞ
Z 1
0
o~p
oyðy;~tÞKðy; ~rÞ dy
þ 4ffiffiffip
pffiffiffiffiffiffiffiffiLð~tÞ
q~KI
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p;
ð26Þ
for the kernel function K given by:
Kðy; ~rÞ ¼ y E arcsin yð Þ ~r2
y2
����� �
� E arcsin vð Þ ~r2
y2
����� �� �
;
ð27Þ
where:
v ¼ min 1;y
~r
�; ð28Þ
with the function E / mjð Þ denoting the incomplete
elliptic integral of the second kind [29].
While this form of the elasticity operator has not
previously been used in the case of a penny-shaped
fracture, an analogous form of the elasticity equation
for the KGDmodel has been utilized in [24, 25], where
its advantages in numerical computations have been
demonstrated. Notably, the kernel function K exhibits
better behaviour than the weakly singular kernelG (7),
having no singularities for any combination of ~r; yf g.Additionally, Eq. (19) can be easily transformed to
obtain p0 and then substituted into (26), meaning that
the latter can be utilized without the additional step of
deriving the pressure function needed for the classic
form of the operator.
Next the boundary conditions (9), in view of (15),
transform to a single condition:
~wð1;~tÞ ¼ 0; ð29Þ
alongside the initial conditions (10):
~wð~r; 0Þ ¼ w�ðrÞl�
; L0 ¼l0
l�: ð30Þ
The source strength (8) is now defined as:
~Q0ð~tÞ2p
¼ lim~r!0
~r ~wð~r;~tÞ~vð~r;~tÞ: ð31Þ
While combining the above with (19) we obtain the
following relationship:
lim~r!0
~rno~p
o~r¼ �
~Q0ð~tÞ2p
� �nLð~tÞ
~w2nþ1ð0; ~tÞ ;ð32Þ
which provides a valuable insight into how the
behaviour of the fluid pressure function near to the
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Meccanica
source varies for differing values of n. The resulting
pressure asymptotics at the injection point, with
corresponding aperture, are detailed below:
~pð~r; ~tÞ ¼~po0ð~tÞ þ ~po1ð~tÞ~r1�n þ O ~r2�n�
; ~r ! 0;
ð33Þ
~wð~r;~tÞ ¼ ~wo0ð~tÞ þ ~wo
1ð~tÞ~r2�n þ O ~r2 logð~rÞ�
; ~r ! 0:
ð34Þ
It is worth restating that there are minor differences to
both the asymptotics and fundamental equations in the
limiting cases n ¼ 0 and n ¼ 1. These are explained in
further detail in ‘‘Appendix 1’’.
3 Crack tip asymptotics, the speed equation
and proper variables
A universal algorithm for numerically simulating
hydraulic fractures has recently been introduced in
[24, 25] and tested against the PKN and KGD (plane
strain) models for Newtonian and shear-thinning
fluids. It proved to be extremely efficient and accurate.
Its modular architecture enables one to adapt it to other
HFmodels by simple replacement or adjustment of the
basic blocks. In the following we will construct a
computational scheme for the radial fracture based on
the universal algorithm. To this end we need to
introduce appropriate computational variables, and to
define the basic asymptotic interrelations between
them. For the sake of completeness detailed informa-
tion on the solutions tip asymptotic behaviour, for
different regimes of crack propagation, are presented
below.
3.1 Crack tip asymptotics
3.1.1 Viscosity dominated regime ( ~KIc ¼ 0)
In the viscosity dominated regime the crack tip
asymptotics of the aperture and pressure derivative
can be expressed as follows:
~wð~r;~tÞ ¼ ~w0ð~tÞ 1� ~r2� a0þ ~w1ð~tÞ 1� ~r2
� a1þ ~w2ð~tÞ 1� ~r2
� a2þ O 1� ~r2
� a2þd �
; ~r ! 1;
ð35Þ
o~p
o~rð~r; ~tÞ ¼ ~p0ð~tÞ 1� ~r2
� a0�2þ ~p1ð~tÞ 1� ~r2� a0�1
þ O 1ð Þ; ~r ! 1: ð36Þ
The crack tip asymptotics of the pressure function can
be derived from the above, however this form is given
due to its use in computations (this will be explained in
further detail later).
As a consequence the fluid velocity behaves as:
~vð~r;~tÞ ¼ ~v0ð~tÞ þ ~v1 ~tð Þ 1� ~r2� b1þO 1� ~r2
� b2 �;
~r ! 1: ð37Þ
Note that we require ~v0ð~tÞ[ 0 to ensure the fracture is
moving forward. The values of constants ai, bi aregiven in Table 1. The general formulae for the limiting
cases n ¼ 0 and n ¼ 1 remain the same as (35)-(37),
with the respective powers ai, bi again being deter-
mined according to Table 1.
Now, let us adopt the following notation for the
crack propagation speed, based on the speed equation
(20) and the tip asymptotics (37):
~v0ð~tÞ ¼ L0ð~tÞ ¼ CLð ~wÞL2ð~tÞ
� �1n
: ð38Þ
Here Lð ~wÞ[ 0 is a known functional and C is a
positive constant. In the viscosity dominated regime
we have that:
C ¼ 2n
ðnþ 2Þ2cot
npnþ 2
� �; Lð ~wÞ ¼ ~wnþ2
0 : ð39Þ
Additionally, we can directly integrate (38) in order to
obtain an expression for the fracture length:
Lð~tÞ ¼ L1þ2nð0Þ þ 1þ 2
n
� �C1
n
Z ~t
0
L1nð ~wÞ ds
" # nnþ2
:
ð40Þ
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Meccanica
3.1.2 Toughness dominated regime ( ~KIc [ 0)
Near the fracture front the forms of the aperture and
fluid velocity asymptotics remain the same as in the
viscosity dominated regime (35), (37). Meanwhile the
pressure derivative asymptotics yields:
o~p
o~rð~r; ~tÞ ¼ ~p0 1� ~r2
� a1�2þ ~p1 1� ~r2� a2�2þO 1ð Þ;
~r ! 1: ð41Þ
The values of ai, bi for this regime are provided in
Table 1. The limiting cases n ¼ 0 and n ¼ 1 are
discussed in ‘‘Appendix 1’’ (Eqs. 78, 87 respectively).
We again use notation (38) for the crack propaga-
tion speed, however the values of the functional L and
the C will in this case be:
C ¼ ð3� nÞð1� nÞ4
tannp2
�; Lð ~wÞ ¼ ~wnþ1
0 ~w1;
ð42Þ
while the fracture length will be given by (40).
3.2 Reformulation in terms of computational
variables
It is readily apparent that the choice of computational
variables plays a decisive role in ensuring the accuracy
and efficiency of the computational algorithm
[23, 24, 26]. Let us introduce a new system of proper
variables which are conducive to robust numerical
computing.
• The reduced fluid velocity Uð~r;~tÞ:
Uð~r;~tÞ ¼ ~r~vð~r; ~tÞ � ~r2~v0ð~tÞ: ð43Þ
It is a smooth, well behaved and non-singular
variable that facilitates the numerical computa-
tions immensely. It is bounded at the crack tip and
the fracture origin. The advantages of using an
analogous variable in the PKN and KGD models
have previously been demonstrated in [24, 25].
• The modified fluid pressure derivative Xð~r; ~tÞ:
~rnXð~r; ~tÞ ¼~rno~p
o~r� X0ð~tÞ; ð44Þ
X0ð~tÞ ¼ �~Q0ð~tÞ2p
� �nLð~tÞ
~w2nþ1ð0; ~tÞ :ð45Þ
It reflects the singular tip behavior of ~p0~r, having
the same tip asymptotics as the pressure derivative,
however it is bounded at the fracture origin. From
(44) the pressure can be immediately reconstructed
as:
~pð~r; ~tÞ ¼ X0ð~tÞ1� n
~r1�n þ Cpð~tÞ þZ ~r
0
Xðn;~tÞdn;
ð46Þ
where the term Cp follows from (25):
Cpð~tÞ ¼1
2
ffiffiffiffiffiffiffiffip
Lð~tÞ
r~KI �
ffiffiffip
pC 3�n
2
� 2 1� nð ÞC 2� n
2
� X0ð~tÞ
�Z 1
0
Xðy; ~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy:
ð47Þ
This auxiliary variable will primarily be used in
numerical computation of the elasticity operator.
The following interrelationship exists between the
newly introduced variables:
Xð~r; ~tÞ ¼~Q0ð~tÞ2p~r
� �nLð~tÞ
~w2nþ1ð0;~tÞ �Lð~tÞ
~wnþ1ð~r; ~tÞUð~r;~tÞ
~rþ ~r~v0ð~tÞ
� �n:
ð48Þ
Since under this new scheme U is bounded at the
fracture tip, the source strength (31) and the boundary
condition (29) can now be expressed as:
~wð0;~tÞUð0;~tÞ ¼~Q0ð~tÞ2p
; ~wð1; ~tÞ ¼ 0: ð49Þ
By utilizing the boundary condition (49)1, the rela-
tionship between the new variables (48) can be
represented in the form:
Table 1 Values of the basic constants used in the asymptotic expansions for ~w and ~v for 0\n\1
Crack propagation regime a0 a1 a2 b1 b2
Viscosity dominated 2
nþ 2
nþ 4
nþ 2
2nþ 6
nþ 2
1 2nþ 2
nþ 2
Toughness dominated 1
2
3� n
2
5� 2n
2
2� n
2
1
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Meccanica
X ~r;~tð Þ ¼ 1
~rnUnð0; ~tÞ~wnþ1ð0; ~tÞ �
Uð~r; ~tÞ þ ~r2~v0ð~tÞð Þn
~wnþ1ð~r; ~tÞ
� �:
ð50Þ
Note that this is not only a more concise representation
of (48) but it also does not depend on Lð~tÞ, which will
be beneficial when analyzing the self-similar formu-
lation. In this way the computational scheme will be
based on: the crack opening, ~w, the reduced fluid
velocity, U, and an auxiliary variable, the modified
fluid pressure, X.By substituting the new variable U from (43) into
the continuity Eq. (18), we obtain the modified
governing equation:
o ~w
o~tþ 1
Lð~tÞ~ro
o~r~wUð Þ þ 2~v0
Lð~tÞ ~wþ ~ql ¼ 0; 0\~r\1:
ð51Þ
Additionally, the elasticity Eq. (26) can be now
expressed as follows:
~wð~r;~tÞ ¼ 8
pLð~tÞ
Z 1
0
Xðy;~tÞKðy; ~rÞ dyþ 4ffiffiffip
pffiffiffiffiffiffiffiffiLð~tÞ
q~KI
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
pþ 8
pLð~tÞX0ð~tÞGnð~rÞ;
ð52Þ
where K is given in (27), while Gn is defined by:
Gnð~rÞ ¼ffiffiffip
pC 3�n
2
� 2 n� 1ð ÞC 2� n
2
�
ffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
pþ 2F1
12; n�2
2; n2; ~r2
� n� 2
�ffiffiffip
p~r2�nC n
2� 1
� 2C n�1
2
� " #
:
ð53Þ
It can be easily shown that this function is well
behaved in the limits.
4 Self-similar formulation
In this section we will reduce the problem to its time-
independent self-similar version. For this formulation
we will define the computational scheme used later on
in the numerical analysis.
We begin by assuming that a solution to the
problem can be expressed in the following manner:
~wð~r;~tÞ ¼ Wð~tÞwð~rÞ; ~pð~r;~tÞ ¼ Wð~tÞLð~tÞ pð~rÞ;
~qð~r; ~tÞ ¼ W2þ2nð~tÞ
L2nð~tÞ
qð~rÞ; ~Q0ð~tÞ ¼W2þ2
nð~tÞL
2nð~tÞ
Q0;
~vð~r; ~tÞ ¼ W1þ2nð~tÞ
L2nð~tÞ
vð~rÞ; Uð~r;~tÞ ¼ W1þ2nð~tÞ
L2n
Uð~rÞ;
~KIð~tÞ ¼Wð~tÞffiffiffiffiffiffiffiffiLð~tÞ
p KI ; Xð~r; ~tÞ ¼ Wð~tÞLð~tÞ Xð~rÞ;
X0ð~tÞ ¼Wð~tÞLð~tÞ X0; Cpð~tÞ ¼
Wð~tÞLð~tÞ Cp;
ð54Þ
where WðtÞ is a smooth continuous function of time.
Such a separation of variables enables one to reduce
the problem to a time-independent formulation when
W is described by an exponential or a power-law type
function. From here on the spatial components will be
marked by a ’hat’-symbol, and will describe the self-
similar quantities. It is worth noting that the separation
of spatial and temporal components given in (54)
ensures that the qualitative behaviour of the solution
tip asymptotics remains the same as in the time-
dependent variant.
4.1 The self-similar representation of the problem
We wish to examine two variants of the time
dependent function:
W1ð~tÞ ¼ ec~t; W2ð~tÞ ¼ aþ ~tð Þc: ð55Þ
In both cases the fluid leak-off function will be
assumed to take the form:
~qlð~r;~tÞ ¼1
cW0ð~tÞqlð~rÞ: ð56Þ
The self-similar reduced fluid velocity (43), modified
fluid pressure derivative (44), (45) and pressure (46)
are defined by:
Uð~rÞ ¼ ~rvð~rÞ � ~r2v0; ~rXð~rÞ ¼ ~rdp
d~r� X0; ð57Þ
pð~rÞ ¼ X0
1� n~r1�n þ Cp þ
Z ~r
0
XðnÞdn; ð58Þ
with
123
Meccanica
X0 ¼� Q0
2p
!n1
w2nþ1ð0Þ ; ð59Þ
Cp ¼ffiffiffip
p
2KI �
ffiffiffip
pC 3�n
2
� 2 1� nð ÞC 2� n
2
� X0 �Z 1
0
XðyÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy:
ð60Þ
It is immediately apparent from (38) and (54) that the
self-similar crack propagation speed is given by:
v0 ¼ lim~r!1
�wnþ1 dp
d~r
� �1n
¼ CLðwÞð Þ1n: ð61Þ
Note again that the qualitative asymptotic behaviour
of the aperture, pressure and fluid velocity as ~r ! 0
and ~r ! 1 remains the same as in the time dependent
version of the problem (35), (36), (37), (41). In the
self-similar formulation, the multipliers of respective
terms are time-independent.
The self-similar counterparts of the elasticity
Eqs. (22) and (23) are now:
pð~rÞ ¼ A½w�ð~rÞ; ð62Þ
where
A½w�ð~rÞ ¼ �Z 1
0
dwðgÞdg
M ~r; g½ � dg; ð63Þ
with the inverse relation being:
wð~rÞ ¼ 8
p
Z 1
0
XðyÞKðy; ~rÞ dyþ 4ffiffiffip
p KI
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p
þ 8
pX0Gnð~rÞ: ð64Þ
As the integral and function Gnð~rÞ both tend to zero
faster than the square root term at the fracture tip, it
immediately follows that, in the toughness dominated
case (KIc [ 0), the leading asymptotic term of the
aperture (35) is given by:
w0 ¼4ffiffiffip
p KI : ð65Þ
The self-similar fluid velocity takes the form:
vð~rÞ ¼ �wnþ1ð~rÞ dpð~rÞd~r
� �1n
: ð66Þ
The governing Eq. (51) becomes:
1
~rv0
d
d~rwU�
¼ � 3� qð Þw� 1� qð Þ qlc; ð67Þ
with the value of q in each case, alongside the fracture
length, provided in Table 2. Meanwhile the fluid
balance condition (21) becomes:
3� qð ÞZ 1
0
~rwð~rÞ d~r þ 1� qc
Z 1
0
~rql d~r ¼Q0
2pv0:
ð68Þ
The self-similar stress intensity factor (25) is given
by:
KI ¼ KIc ¼2ffiffiffip
pZ 1
0
~rpð~rÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p d~r: ð69Þ
Finally, the system’s boundary conditions (49) trans-
form to:
wð0ÞUð0Þ ¼ Q0
2p; wð1Þ ¼ 0: ð70Þ
In the general case with 0\n\1 these equations
represent the full self-similar problem. Some modifi-
cations are necessary in the special cases when n ¼ 0
and n ¼ 1. These differences are outlined in ‘‘Ap-
pendix 1’’.
5 Numerical results
In this section we will construct an iterative compu-
tational scheme for numerically simulating hydraulic
fracture. The approach is an extension of the universal
algorithm introduced in [24, 25]. The computations
are divided between two basic blocks, the first of
which utilizes the continuity equation and the latter
using the elasticity operator. The previously intro-
duced computational variables, alongside the known
Table 2 Table providing the fracture length Lð~tÞ, which is
obtained using (40) and (61), and the constant q, used in (67)
and (68), for different variants of the self-similar solution
Self-similar law q Lð~tÞ
Wð~tÞ ¼ ec~t 0 v0c
h i nnþ2
ec~t
Wð~tÞ ¼ aþ ~tð Þc nc nþ2ð Þþn nþ2ð Þv0
c nþ2ð Þþn
h i nnþ2
aþ ~tð Þcþn
nþ2
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Meccanica
information about the solution tip asymptotics, are
employed extensively. The accuracy and efficiency of
the computations are verified against the newly
introduced analytical benchmark examples. Then the
numerical benchmark solutions are given. Finally, a
comparative analysis with other data available in the
literature is delivered.
5.1 Computational scheme
The algorithm is constructed using the approach
framework introduced for the PKN and KGD models
in [24, 25]. The numerical scheme is realized as
follows:
1. An initial approximation of the aperture w ¼ wj�1
is taken, such that it has the correct asymptotic
behaviour and satisfies the boundary conditions.
2. The fluid balance Eq. (68) is utilized to obtain the
asymptotic term(s) wj0;1 needed to compute the
fluid velocity vj0 using (61).
3. Having the above values the reduced fluid velocity
U j is reconstructed by direct integration of (67).
Tikhonov type regularization is employed at this
stage.
4. Equations (57) and (66) is then used to obtain an
approximation of the modified fluid pressure
derivative X, and the elasticity Eq. (64) serves to
compute the next approximation of the fracture
aperture wj.
5. The system is iterated until all variables U, w and
v0 converge to within prescribed tolerances.
We will demonstrate in this section that this scheme,
combined with an appropriate meshing strategy, yields
a highly accurate algorithm. A more detailed descrip-
tion of the algorithm’s construction has been outlined
in [24, 25]. When iterated, the system of discretized
equations converges to a final solution in under 20
iterations, with computation times on a standard laptop
under 5 s when taking N ¼ 20, and under 30 s when
N ¼ 300.
It is worth noting that, due to the degeneration of the
Poiseuille equation when n ¼ 0, it can no longer be
used to compute the fluid flow rate or the fluid
velocity. However, thanks to the modular structure of
the proposed algorithm, one can easily adapt it to this
variant of the problem. In this case a special form of
the elasticity Eq. (96) is utilized to obtain the aperture,
with the fluid velocity being reconstructed using
relations (97) and (98).
5.2 Accuracy of computations
In this subsection we will investigate the accuracy of
computations delivered by the proposed numerical
scheme. To this end a newly introduced set of
analytical benchmark solutions with a non-zero fluid
leak-off function will be used. Alternative measures
for testing the numerical accuracy in the absence of
exact solutions will then be proposed and analysed.
Next, the problem of a penny-shaped hydraulic
fracture propagating in an impermeable material will
be considered. The accuracy of numerical solutions
will be verified by the aforementioned alternative
measures. Simple, semi-analytical approximations,
which mimic the obtained numerical data to a
prescribed level of accuracy will be provided in the
second part of this paper, alongside a comparative
analysis with other solutions available in the literature.
5.2.1 Analysis of computational errors against
analytical benchmarks
The first method of testing the computational accuracy
is by comparison with analytical benchmark solutions.
Respective closed-form benchmarks with predefined,
non-zero, leak-off functions are outlined in the
supplementary material associated with this paper.
They have been constructed for both the viscosity and
toughness dominated regimes, for a class of shear-
thinning and Newtonian fluids. All of the analytical
benchmarks used for comparison are designed to
ensure physically realistic behaviour of the solution
while maintaining the proper asymptotic behaviour. In
all numerical simulations the power-law form of the
time dependent function W2 (55)2 is used to ensure
consistency with results in the literature (e.g. [19]).
The accuracy of computations is depicted in Figs. 1
and 2, for varying number of nodal points N. A non-
uniform spatial mesh was used, with meshing density
increased near the ends of the interval (the same type
of mesh was used for all n). The measures dw, dv,describing the average relative error of the crack
opening and fluid velocity, are taken to be:
123
Meccanica
dwðNÞ ¼R 10~r w�ð~rÞ � wð~rÞj j d~rR 1
0~rw�ð~rÞ d~r
;
dvðNÞ ¼R 10~r v�ð~rÞ � vð~rÞj j d~rR 1
0~rv�ð~rÞ; d~r
;
ð71Þ
where w� and v� denote the exact solutions for w and v.
The results clearly show that the values of both
error measures decrease monotonically with growing
N. For a fixed number of nodal points N, dw is lower
than dv, but within the same order of magnitude. One
can observe a sensitivity of the results to the value of
the fluid behaviour index n. Here, the level of error
measures can vary up to one order for a constant N.
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)δw
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
(b)
Fig. 1 Relative average error of the crack aperture (71)1 obtained against the analytical benchmark overN for the a viscosity dominated
regime, b toughness dominated regime
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)
δv
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
(b)
Fig. 2 Relative average error of the fluid velocity (71)2 obtained against the analytical benchmark over N for the a viscosity dominated
regime, b toughness dominated regime
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Meccanica
This trend can be alleviated by adjusting the mesh
density distribution to the value of n (i.e. to the varying
asymptotics of solution), however such an investiga-
tion goes beyond the scope of this paper. In general, it
takes fewer than N ¼ 300 nodal points to achieve the
relative errors of the level 10�7.
In cases when the exact solution is not prescribed an
alternative method of testing the solution accuracy is
required. The method outlined here relies on the fact
that the solution converges to the exact value at a
known rate, with respect to the number of nodal points.
The convergence rate has been established numeri-
cally to behave as 1=N3. As a result the following
estimation holds:
Z 1
0
~rgið~rÞ d~r ¼ Ai þBi
N3; i ¼ 1; 2; ð72Þ
where g1ð~rÞ ¼ wð~rÞ and g2ð~rÞ ¼ vð~rÞ. Ai and Bi are
constants which can be found numerically through use
of a least-squares approximation. Next, one can define
the limiting value of (72) as:
limN!1
Z 1
0
~rgið~rÞ d~r ¼ Ai Z 1
0
~rg�i ð~rÞ d~r; i ¼ 1; 2;
ð73Þ
for g�1ð~rÞ ¼ w�ð~rÞ, g�2ð~rÞ ¼ v�ð~rÞ.Knowing this, the following alternative error mea-
sures can be proposed:
egiðNÞ ¼ 1� 1
Ai
Z 1
0
~rgið~rÞ d~r����
����; i ¼ 1; 2: ð74Þ
Using this strategy, it is possible to identify the relative
rate at which the solution converges: ewðNÞ for the
aperture and evðNÞ for the fluid velocity. The results
are shown in Figs. 3 and 4. It is notable that both dwand ew, as well as dv and ev, provide estimates of a
similar order for a fixed N. Thus, both of these error
measures can be employed in the accuracy analysis,
although only the latter (ew;v) in the cases when no
exact solution is available. As such, ewðNÞ and evðNÞwill be used in the following investigations (it should
also be noted that this approach has previously been
shown to be successful for the PKN/KGD models of
HF [24, 25, 33]).
5.2.2 Impermeable solid: reference solutions
With a suitable measure for testing the solution
accuracy in place we move onto examining the
solution variant most frequently studied in the liter-
ature, the case with a zero valued leak-off function and
with Q0 ¼ 1. Although there is no analytical solution
to this problem, due to its relative simplicity, it is
commonly used when testing numerical algorithms.
For this reason it is very important that credible
reference data is provided for this case, which can be
easily employed to verify various computational
schemes. Both the viscosity and toughness dominated
regimes (for different values of the material tough-
ness: KIc ¼ 1; 10; 100f g) will be investigated. In the
second part of the paper, accurate and simple approx-
imations of the obtained numerical solutions will be
provided.
The results for the crack opening and fluid velocity
convergence rates are shown in Figs. 5, 6, 7 and 8.
As can be seen, over the analyzed range of N, the
computations are very accurate and converge rapidly
as the mesh density is increased. In the viscosity
dominated regime it can be seen that there is a lower
sensitivity of ew and ev to the value of n, however even
in the toughness dominated mode the dependence of
ew on the fluid behaviour index becomes less pro-
nounced as KIc grows. A general trend can be
observed, in that the convergence rate is magnified
as the self-similar material toughness KIc increases.
This is due to the fact that, for large values of KIc, the
solution tends to the limiting case of a uniformly
pressurized immobile crack with a parabolic profile.
To explain this tendency we present in Figs. 9, 10, 11
and 12 some additional data for a single value of the
fluid behavior index (n ¼ 0:5).
It is immediately obvious that for KIc [ 2 the
fracture aperture is almost entirely described by the
leading term of its crack tip asymptotics (for KIc ¼ 2
the maximal deviation between them is approximately
1 percent). For the fluid velocity it can be seen that,
while the effect is not as substantial as for the aperture,
the crack propagation speed v0 does become a better
predictor of the parameter’s behaviour for larger
values of the material toughness. Meanwhile, the fluid
pressure increases with growing KIc, eventually
becoming uniformly distributed over ~r. As a result of
the decreasing pressure gradient the velocity of the
123
Meccanica
fluid flow is reduced. In Fig. 12 it can be seen that the
fluid flow rate rapidly converges to the limiting case
with growing KIc, however the rate of convergence is
greater for larger values of n. Indeed, as can be seen in
Fig. 13, for n ¼ 1 the curves for KIc ¼ 1 and KIc ¼100 are indistinguishable, which is not the case when
n ¼ 0.
In fact, the behaviour of the solution as KIc ! 1can easily be shown to take the form:
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
N
n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)e w
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
N
(b)
Fig. 3 Rate of convergence ew (74) of the numerical solution for the benchmark example: a viscosity dominated regime, b toughness
dominated regime
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)
e v
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
(b)
Fig. 4 Rate of convergence ev (74) of the numerical solution for the benchmark example: a viscosity dominated regime, b toughness
dominated regime
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Meccanica
wð~rÞ� 4ffiffiffip
p KI
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p; pð~rÞ�
ffiffiffip
p
2KI ;
v0 �3
8ffiffiffip
pKIð3� qÞ
;
ð75Þ~rvð~rÞ ¼v0 ~r2 þ 3� q
31� ~r2� � �
þ O K�1Ic
� ; ð76Þ
~rqð~rÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p
2p3~r2
3� qþ 1� ~r2� � �
þ O K�1Ic
� ;
ð77Þ
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
n=0n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)e w
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
(b)
Fig. 5 Rate of convergence ew (74) of the numerical solution whenQ0 ¼ 1 with no fluid leak-off for the: a viscosity dominated regime,
b toughness dominated regime with KIc ¼ 1
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
n=0n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)
e w
50 100 150 200 250 30010
−9
10−8
10−7
10−6
10−5
10−4
N
(b)
Fig. 6 Rate of convergence ew (74) of the numerical solution when Q0 ¼ 1 with no fluid leak-off for the toughness dominated regime
with: a KIc ¼ 10 and b KIc ¼ 100
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Meccanica
where q is defined in Table 2. As a result the
computations become far more efficient in this case
and the resulting solution is calculated to a far higher
level of accuracy.
Combining the results shown above in Figs. 1, 2, 3,
4, 5, 6, 7 and 8, it is clear that the computations
presented here achieve a very high level of accuracy
for both the aperture and fluid velocity regardless of
the crack propagation regime. When using N ¼ 300
the accuracy of computations can almost always be
assumed to be correct to a level of at least 10�7 for the
fracture aperture, and 2:5 10�7 for the fluid velocity.
In this way the obtained data constitutes a very
convenient and credible reference solution when
50 100 150 200 250 30010
−12
10−10
10−8
10−6
10−4
N
n=0n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)e v
50 100 150 200 250 30010
−12
10−10
10−8
10−6
10−4
N
(b)
Fig. 7 Rate of convergence ev (74) of the numerical solution whenQ0 ¼ 1 with no fluid leak-off for the: a viscosity dominated regime,
b toughness dominated regime with KIc ¼ 1
50 100 150 200 250 30010
−12
10−10
10−8
10−6
10−4
N
n=0n=0.1n=0.2n=0.3n=0.4n=0.5n=0.6n=0.7n=0.8n=0.9n=1
(a)
e v
50 100 150 200 250 30010
−12
10−10
10−8
10−6
10−4
N
(b)
Fig. 8 Rate of convergence ev (74) of the numerical solution when Q0 ¼ 1 with no fluid leak-off for the toughness dominated regime
with: a KIc ¼ 10 and b KIc ¼ 100
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Meccanica
testing other computational schemes. Simple (based
on elementary functions) and accurate approximations
of the results, which facilitates their application as
benchmark data, are provided in the second part of this
paper.
It is worth mentioning that the efficiency of
computations achieved by this algorithm means that
this high level of accuracy does not come at the
expense of simulation time. The final algorithm
requires fewer than 20 iterations to produce a solution.
Simulation times are also very short with this scheme.
6 Conclusions
In this paper, the problem of a penny-shaped hydraulic
fracture driven by a power-law fluid has been
analyzed. Following an approach similar to that in
[24, 25] the governing equations have been
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r
KIc = 0KIc = 1KIc = 2KIc = 5KIc = 10KIc = 100
(a)w(r)
w(0)
0 0.2 0.4 0.6 0.8 1
0.98
1
1.02
1.04
1.06
1.08
1.1
r
(b)
w(r)
w0(1−r
2)α
0
Fig. 9 The aperture for n ¼ 0:5 for a different values of the fracture toughness: a the normalized self-similar aperture, b the self-similar
aperture divided by the leading term of the crack tip asymptotics (35)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
r
KIc = 0KIc = 1KIc = 2KIc = 5KIc = 10KIc = 100
(a)
rv(r)
0 0.2 0.4 0.6 0.8 1
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
r
(b)rv(r)
v0
Fig. 10 The fluid velocity for n ¼ 0:5 for different values of the fracture toughness: a the self-similar fluid velocity, b the self-similar
fluid velocity divided by the leading term of the crack tip asymptotics (37)
123
Meccanica
reformulated in terms of the aperture w and the
reduced fluid velocity U. Self-similar formulations
have been derived for two types of time dependent
function. A computational scheme based on the
universal algorithm introduced in [24] has been
constructed. The accuracy of computations has been
verified against a set of newly introduced analytical
benchmark examples. Alternative measures of the
solution accuracy have been proposed and investi-
gated. The ability to obtain highly accurate numerical
reference solutions has been demonstrated.
The following conclusions can be drawn from the
conducted research.
• The universal algorithm for numerically simulat-
ing hydraulic fractures, introduced in [24], can be
successfully adapted to the case of a penny-shaped
fracture. It enables accurate and efficient mod-
elling of HFs driven by the power-law fluids in
both the viscosity and toughness dominated
regimes.
• The key elements of the algorithm, which con-
tributed to its outstanding performance, are: i)
choice of proper computational variables, includ-
ing the reduced fluid velocity, ii) extensive
utilization of the information on the solution
asymptotics, combined with a fracture front trac-
ing mechanism based on the Stefan-type condition
(speed equation), iii) application of the modified
form of the elasticity operator (26), which has a
non-singular kernel, that can easily be coupled
with the new dependent variable - the reduced fluid
velocity.
• The newly introduced analytical benchmark solu-
tions, with a predefined non-zero fluid leak-off,
can be adjusted to mimic the HF behaviour for a
class of power-law fluids in both the viscosity and
toughness dominated regimes. These solutions can
be directly applied to investigate the actual error of
computations when testing various computational
schemes.
• The error measures ew and ev (74), based on the
rate of solution convergence, have been shown to
0 0.2 0.4 0.6 0.8 110
−3
10−2
10−1
100
101
102
r
KIc = 0KIc = 1KIc = 2KIc = 5KIc = 10KIc = 100
(a)p(r)
0 0.2 0.4 0.6 0.8 1−8
−6
−4
−2
0
2
r
(b)
p(r)
p(0)
Fig. 11 The pressure function for n ¼ 0:5 for different values of the fracture toughness: a the self-similar pressure function, b the self-
similar pressure divided by the value of the pressure at the fracture opening
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r
KIc = 0KIc = 1KIc = 2KIc = 5KIc = 10KIc = 100
2πrq
Fig. 12 The self-similar fluid flow rate for n ¼ 0:5 for differentvalues of the fracture toughness
123
Meccanica
be equivalent and credible error measures for
analyzing the problem when no closed-form
analytical solutions are available.
While this work has allowed for the creation of highly
accurate numerical benchmarks, they are not in a form
which can be easily utilized. In the second part of this
paper, approximate formulae for the case of an
impermeable solid, constituting a set of accurate and
easily accessible reference solutions when investigat-
ing other computational algorithms, will be delivered.
Additionally, a brief comparison with alternative
benchmarks available in the literature will be
performed.
Acknowledgements All authors are grateful to the funding
bodies who supported this project. DP andMW are very grateful
to ISOTOP for the facilities they provided during their
secondments. Both would specifically like to thank Dr Vladi
Frid for his fruitful discussions when beginning the paper, and
throughout their secondment periods. GM is grateful to the
Royal Society for the Wolfson Research Merit Award.
Funding DP was funded by the European Union Seventh
Framework Marie Curie IAPP project PARM-2 (reference:
PIAP-GA-2012-284544) and H2020 Marie Sklodowska Curie
RISE project MATRIXASSAY (H2020-MSCA-RISE-2014-
644175). MW received funding from the FP7 PEOPLE Marie
Curie IRSES project TAMER (reference: IRSES-GA-2013-
610547) and acknowledges support from the project First
TEAM/2016-1/3 of the Foundation for Polish Science, co-
financed by the European Union under the European Regional
Development Fund. GM gratefully acknowledges partial
supports from the ERC Advanced Grant ‘‘Instabilities and
nonlocal multiscale modeling of materials ERC-2013-ADG-
340561-INSTABILITIES during his Visiting Professorship at
Trento University and the Grant No. 14.581.21.0027 unique
identifier: RFMEFI58117X0027 by Ministry of Education and
Science of the Russian Federation. MP is supported by the FP7
PEOPLE Marie Curie action project CERMAT2 under number
PITN-GA-2013-606878.
Compliance with ethical standards
Conflict of interest The authors declare that these publicly
funded arrangements haven’t created a conflict of interest.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unre-
stricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original
author(s) and the source, provide a link to the Creative Com-
mons license, and indicate if changes were made.
Appendix 1: Limiting cases: Newtonian and plastic
fluids
Newtonian fluid: n ¼ 1
In the case of a Newtonian fluid the majority of the
results remains the same as in the general case (setting
n ¼ 1), but a few constants and functions will take
alternate forms. These are detailed below.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r
KIc = 0KIc = 1KIc = 2KIc = 5KIc = 10KIc = 100
(a)2π
rq
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r
(b)
2πrq
Fig. 13 The self-similar fluid flow rate for different values of the fracture toughness when the fluid behaviour index is: a n ¼ 0 and
b n ¼ 1
123
Meccanica
The crack tip asymptotics in the viscosity domi-
nated regime can be described by general relations
(35)–(37). However, in the toughness dominated mode
one has:
~wð~r;~tÞ ¼ ~w0ð~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
pþ ~w1ð~tÞ 1� ~r2
� þ ~w2ð~tÞ 1� ~r2
� 32log 1� ~r2
� þ O 1� ~r2
� 32
�; ~r ! 1;
ð78Þ
o~p
o~r¼ ~p0ð~tÞ 1� ~r2
� �1þ~p1ð~tÞ 1� ~r2� �1
2þO 1ð Þ; ~r ! 1:
ð79Þ
The respective asymptotic expansions at the crack
inlet, for both the viscosity and toughness dominated
regimes, yield:
~wð~r;~tÞ ¼ ~wo0 þ ~wo
1~r þ O ~r2 logð~rÞ�
; ~r ! 0; ð80Þ
~pð~r; ~tÞ ¼~po0ð~tÞ þ ~po1ð~tÞ log ~rð Þ þ O ~rð Þ; ~r ! 0:
ð81Þ
It should be noted that the pressure is singular at the
fracture origin, which is not the case for non-Newto-
nian (n\1) fluids.
Meanwhile, the relationship between the new
variable X and the pressure, in the time-dependent
formulation, follows from the definition (44):
~pð~r; ~tÞ ¼ X0ð~tÞ logð~rÞ þ Cpð~tÞ þZ ~r
0
Xðn; ~tÞdn;
ð82Þ
where the time dependent constant Cpð~tÞ is obtainedby expanding (25) using (44):
Cpð~tÞ ¼1
2
ffiffiffiffiffiffiffiffip
Lð~tÞ
r~KI þ 1� log 2ð Þ½ �X0ð~tÞ
�Z 1
0
Xðy; ~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy:
ð83Þ
Transforming into the self-similar formulation (54),
these become:
pð~rÞ ¼ X0 log ~rð Þ þ Cp þZ ~r
0
XðnÞ dn; ð84Þ
Cp ¼ffiffiffip
p
2KI þ 1� log 2ð Þ½ �X0 �
Z 1
0
XðyÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy:
ð85Þ
Finally, the auxiliary function Gnð~rÞ will now be
expressed as:
Gnð~rÞ ¼ ~rp2� arctan
~rffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p� �� �
�ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p� ~r arccos ~rð Þ �
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p:
ð86Þ
Perfectly plastic fluid: n ¼ 0
In the case of a perfectly plastic fluid, alongside
changes to the system asymptotics and reformulated
equations, the degeneration of the Poiseuille equation
means that it cannot be used to define the fluid velocity
~v, or the reduced fluid velocity U. As a result
fundamental changes to the scheme are required.
These are outlined below.
The crack tip asymptotics in the viscosity domi-
nated regime remains in the same form as was outlined
in (35)–(37). In the toughness dominated mode
however it now yields:
~wð~r;~tÞ ¼ ~w0ð~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
pþ ~w1ð~tÞ 1� ~r2
� 32log 1� ~r2
� þ ~w2ð~tÞ 1� ~r2
� 32þO 1� ~r2
� 52
�; ~r ! 1;
ð87Þ
o~p
o~r¼ ~p0ð~tÞ 1� ~r2
� �12þO 1ð Þ; ~r ! 1: ð88Þ
The fracture opening and the fluid pressure can be
estimated at the crack inlet as:
~wð~r;~tÞ ¼ ~wo0ð~tÞ þ O ~r2 logð~rÞ
� ; ~r ! 0; ð89Þ
~pð~r; ~tÞ ¼~po0ð~tÞ þ ~po1ð~tÞ~r þ O ~r2�
; ~r ! 0: ð90Þ
Meanwhile, the relationship between the modified
fluid pressure derivative and the pressure follows from
the definition (44):
~pð~r; ~tÞ ¼ ~rX0ð~tÞ þ Cpð~tÞ þZ ~r
0
Xðn;~tÞ dn; ð91Þ
where
123
Meccanica
Cp ¼1
2
ffiffiffiffiffiffiffiffip
Lð~tÞ
r~KI �
p4X0ð~tÞ �
Z 1
0
Xðy; ~tÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy:
ð92Þ
Note, from the form of the above, that the pressure is
not singular at the injection point in this case.
Transforming into the self-similar formulation (54)
these become:
pð~rÞ ¼ ~rX0 þ Cp þZ ~r
0
XðnÞ dn; ð93Þ
Cp ¼ffiffiffip
p
2KI �
p4X0 �
Z 1
0
XðyÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
pdy: ð94Þ
It can be shown that the relationship betweenX and the
fracture aperture (52) still holds, with the function
Gnð~rÞ being given by:
Gnð~rÞ ¼ � p8
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
pþ ~r2 log
~r
1þffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p� �� �
� � p8
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p� ~r2arctanh
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p �h i:
ð95Þ
In practice however, the degeneration of the Poiseuille
equation means that a new scheme for solving the
governing equations must be devised. The first step
towards this is to note that the fracture aperture can be
expressed as a non-linear integral equation:
wð~rÞ ¼ � 8
p
Z 1
0
1
wðyÞ Kðy; ~rÞ; dyþ 4ffiffiffip
p KI
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ~r2
p;
ð96Þ
while the crack-propagation speed is calculated from
the fluid balance Eq. (68) as follows:
v0 ¼Q0
2p 3� qð ÞR 10~rwð~rÞ d~r þ 1�q
c
R 10~rql d~r
h i : ð97Þ
The reduced fluid velocity U can be determined by
integrating (67):
Uð~rÞ ¼ v0
wð~rÞ
Z 1
~r
n 3� qð ÞwðnÞ þ 1� qð Þ qlðnÞc
� �dn:
ð98Þ
Relations (96)–(98) are embedded accordingly in the
general numerical scheme.
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